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The problem presented involves a firm using two inputs $x$ and $y$ to maximize profit with a given budget constraint. Let's break it down:

### Key Information:
- **Profit function**: $P(x, y) = 3xy - 2x + y$
- **Cost of inputs**:
  - $x$ costs $2 each.
  - $y$ costs $1 each.
- **Budget constraint**: The firm has $100 to spend on inputs.

We are tasked with finding:
1. The optimal quantities of $x$ and $y$ that maximize the firm's profit.
2. The shadow price (also known as the Lagrange multiplier) associated with this optimization.

### Steps to Solve:

#### 1. Set up the constraint:
The total spending constraint is given by:
\[
2x + y = 100
\]
This is the budget the firm must adhere to.

#### 2. Set up the Lagrange function:
We use the method of **Lagrange multipliers** to solve this problem. The Lagrange function, $\mathcal{L}(x, y, \lambda)$, combines the objective function and the constraint:
\[
\mathcal{L}(x, y, \lambda) = 3xy - 2x + y + \lambda(100 - 2x - y)
\]
where $\lambda$ is the Lagrange multiplier (shadow price).

#### 3. Take partial derivatives:
We need to take the partial derivatives of $\mathcal{L}$ with respect to $x$, $y$, and $\lambda$, and set them equal to zero to find the critical points.

- $\frac{\partial \mathcal{L}}{\partial x} = 3y - 2 - 2\lambda = 0$
- $\frac{\partial \mathcal{L}}{\partial y} = 3x + 1 - \lambda = 0$
- $\frac{\partial \mathcal{L}}{\partial \lambda} = 100 - 2x - y = 0$ (This is just the original constraint.)

#### 4. Solve the system of equations:
From these, we have the system:
1. $3y - 2 = 2\lambda$
2. $3x + 1 = \lambda$
3. $2x + y = 100$

We can solve this system step by step to find the values of $x$, $y$, and $\lambda$.

Let me compute the solution next.The solution to the system is:

- $x = \frac{74}{3} \approx 24.67$
- $y = \frac{152}{3} \approx 50.67$
- The shadow price (Lagrange multiplier) $\lambda = 75$

### Conclusion:
- The firm should use **24.67 units of input $x$** and **50.67 units of input $y$** to maximize profit.
- The **shadow price** is **75**. This indicates the marginal value of increasing the budget constraint by $1, which would result in an additional profit of $75.

Would you like more details on any part of this process?

### Related Questions:
1. What is the interpretation of the shadow price in a different context?
2. How do Lagrange multipliers generalize to more complex constraints?
3. What is the significance of the budget constraint in profit maximization problems?
4. How would the solution change if the cost of $x$ increased?
5. What happens if the budget is doubled while keeping the prices the same?

### Tip:
In constrained optimization, the shadow price provides valuable insight into how sensitive the objective is to changes in the constraint.

A firm uses two inputs x and y, and their profit function is P(x,y) = 3xy - 2x + y. Input x costs $2 each and y costs $1 each, and they are constrained to spend a total of $100 on inputs. If the firm wants to maximize profit, they should use ___ units of input x, ___ units of input y. In addition, the shadow price will be ___. Round your answer to two decimal places where necessary.

Learn how to maximize profit for a firm using two inputs and a given budget constraint. This problem uses Lagrange multipliers to determine the optimal values for input x and input y, given their respective costs. The solution includes finding the shadow price and interpreting its significance.

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